obtain a second linearly independent solution to the differential equation. The factor of 1/2 is inserted in order to make correspond to the Bessel functions of the second kind. We have and In terms of K, we have Hence, just as the Bessel function and modified Bessel function of the first kind can both be expressed in terms of , those of the second kind can both be expressed in terms of . Generating function If we multiply the absolutely convergent series for exp(t) and exp(z/t) together, we get (when t is not zero) an absolutely convergent series for exp(t + z/t). Collecting terms in t, we find on comparison with the power series definition for that we have This generating function can then be used to obtain further formulas, in particular we may use Cauchy's integral formula and obtain for integer n as References . . . . . . Special hypergeometric functions